cp's OEIS Frontend

A344073 Smallest number k such that C(-k) is the cyclic group of order n, where C(D) is the class group of the quadratic field with discriminant D; or 0 if no such k exists.

%I A344073 #17 May 09 2021 02:24:07
%S A344073 3,15,23,39,47,87,71,95,199,119,167,327,191,215,239,407,383,335,311,
%T A344073 776,431,591,647,695,479,551,983,831,887,671,719,791,839,1079,1031,
%U A344073 959,1487,1199,1439,1271,1151,1959,1847,1391,1319,2615,3023,1751,1511,1799
%N A344073 Smallest number k such that C(-k) is the cyclic group of order n, where C(D) is the class group of the quadratic field with discriminant D; or 0 if no such k exists.
%C A344073 Different from A060649.
%C A344073 Conjecture 1: a(n) > 0 for all n;
%C A344073 Conjecture 2: a(n) = o(n^2).
%C A344073 What's the next even term after a(20) = 776 and a(104) = 14024?
%H A344073 Jianing Song, <a href="/A344073/b344073.txt">Table of n, a(n) for n = 1..488</a>
%F A344073 For odd n, if a(n) > 0, then a(n) >= A060649(n). The smallest odd n such that the inequality is strict is n = 243.
%F A344073 For even n, if a(n) > 0, A060649(n) > 0 and A344072(n/2) > 0, then a(n) >= min{A060649(n), A344072(n/2)/4}. Assuming Conjecture 2 in A344072, we have a(n) >= A060649(n). The smallest n == 2 (mod 4) such that the inequality is strict is n = 342.
%e A344073 The smallest k such that c(-k) = C_12 is k = 327, so a(12) = 327.
%e A344073 The smallest k such that c(-k) = C_16 is k = 407, so a(16) = 407.
%e A344073 The smallest k such that c(-k) = C_20 is k = 776, so a(20) = 776.
%e A344073 The smallest k such that c(-k) = C_243 is k = 38231, so a(243) = 38231.
%o A344073 (PARI) a(n) = if(n==1, 3, my(d=3); while(!isfundamental(-d) || quadclassunit(-d)[2]!=[n], d++); d)
%Y A344073 Cf. A060649, A344072.
%K A344073 nonn
%O A344073 1,1
%A A344073 _Jianing Song_, May 08 2021